The Growth of Polarization Domains in Ultrathin Ferroelectric Films Seeded by the Tip of an Atomic Force Microscope

Piezoresponse force microscopy is used to study the velocity of the polarization domain wall in ultrathin ferroelectric barium titanate (BTO) films grown on strontium titanate (STO) substrates by molecular beam epitaxy. The electric field due to the cone of the atomic force microscope tip is demonstrated as the dominant electric field for domain expansion in thin films at lateral distances greater than about one tip diameter away from the tip. The velocity of the domain wall under the applied electric field by the tip in BTO for thin films (less than 40 nm) followed an expanding process given by Merz’s law. The material constants in a fit of the data to Merz’s law for very thin films are reported as about 4.2 KV/cm for the activation field, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E_{\mathrm{a}}$$\end{document}Ea, and 0.05 nm/s for the limiting velocity, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v_{\infty }$$\end{document}v∞. These material constants showed a dependence on the level of strain in the films, but no fundamental dependence on thickness. Supplementary Information The online version contains supplementary material available at 10.1186/s11671-022-03688-2.

Spherical coordinate system was used to approximate the electric field in the ferroelectric thin film between a conductive tip and a conductive substrate at lateral distances greater than about one tip diameter from the tip-surface contact (taken as origin). The geometry and the parameters of the structure were described in Fig. 1. In the approximation for the electric field in the film, the lateral size of the film and the cone is assumed to be infinite.
The spherical part of the tip is neglected when we are considering lateral distances, r, greater than about one tip diameter from the origin. The film is assumed to be thin compared to the lateral distance, and the electric field in the film is also assumed to be uniform in the film. Due to the symmetry in Fig. 1, the three-dimensional Laplace equation for electric potential, φ, in spherical coordinate can be reduced to: The solution for the differential equation is: Where A and B are integration constants. The electric field is in the θ-direction and given by: The electric field in the air and the film are therefore given by:

E air
Here, d is thickness of the film, θ 0 is the cone half angle as shown in the Fig. 1 and E f ilm θ is approximately in the z-direction and equivalent to E f ilm z for thin films. In the case where a thin dielectric layer forms on the surface [2][3][4], the change in the electric field due to the cone is negligible. Since f ilm for BTO is much greater than air , we can neglect E z f ilm in Eq. S7 and find A 2 as Therefore, the magnitude of the electric field due to the cone in the film is approximated as: By putting air = 1 and θ = π/2, the equation for the electric field perpendicular to the film at distances greater than about one tip diameter and larger than thickness of the film is given by: BTO is an anisotropic dielectric [5] with c ≈ 200 and a ≈ 4000 . Since the electric field in the film away from the tip is mostly in z-direction, the c-dielectric constant, c , is taken for f ilm . The cone of the tip is a truncated cone so we need to transform r to r − r 0 that r 0 is the distance of the intercept of the cone with the surface from the origin. By considering these substitutions Eq. S10 results in the Eq. 5 in the paper. The geometry of the tip, including the angle of the cone, θ 0 , the radius of tip apex, a, as well as the intercept of the cone of the tip with the surface, r 0 , are needed to find the applied electric field from the AFM tip.
To find these parameters we used a reference sample (TGT1) patterned with pointed structures having a 10 nm apex radius and 500 nm height which was used to form an AFM image using our AFM tip as shown in Fig. S1. This was then used to deconvolve out the tip parameters shown in Fig. S2 The results for "tip #1 measured" and "tip #2 measured" are compared with the "tip specifications" given by the company in Table S1.

D. DEVELOPMENT OF EQ. 6
To find the equation that describes the domain radius as a function of the poling time, the electric field (Eq. 5) is substituted into Merz's law (Eq. 2), and the velocity of domain wall is written as dr/dt in which r is the domain radius and t the poling time. Therefore, we have: To simplify the calculation we define γ as: Separating r and t terms on the two sides of equation (S11) gives: Now we can integrate both sides with the initial values of (t i , r i ) and final values of (t, r), and the result is: By rearranging Eq. S14 to find r as a function of t, we have Eq. 6.

E. THE ELECTRIC FIELD IN THE FILM FROM COMSOL SIMULATIONS
We used the finite element method of COMSOL Multiphysics to simulate the electric field due to the AFM tip. The AFM tip is defined as a sphere and a truncated cone (Fig. 6S). In the simulation, the cone height is 800nm, the half angle of the cone, 20°, the width of the film and substrate, 1400 nm, the dielectric constants of the barium titanate (BTO) film [5], x = 4000, y = 4000, z = 200 and the dielectric constant of the material around the tip, 1. The tip and substrate were taken as "terminals" with the voltage of 0 V and 7 V respectively.
Since there are reports [2][3][4]on the presence of an ultrathin dielectric surface layer (carbon compounds) on ferroelectric materials, 0.5 nm dielectric layer with the dielectric constant of 2 was considered on the films. We simulated the electric field for 2, 10, and 40 nm films and two different tip apex radii of 30 nm and 60 nm. The mesh was selected based on no change in simulation with selection of a finer mesh size. The simulation shows that the electric field in the z-direction under the tip is larger near the surface of the film (z = 0) and decreasing moving toward the substrate (z = −10 nm).
This is observed in Fig. S7 by comparing the slope of the electric potential vs. z at z = 0, to the slope at z = −10 nm, for a 10 nm film at r = 0 and r = 10 nm. However, the electric field becomes uniform for larger r as shown by the simulated data at r = 100 nm for a 10 nm film in Fig. S7.
The electric field in the z-direction at the surface of the films (z = 0) for different films are compared in Fig. S8 indicating the behavior of the electric field as a function of lateral distance, r, for different thicknesses of the film. The electric field is clearly larger for thinner films than thicker films at r = 0. However, this order changes at r = 50 nm and the electric field becomes bigger for thicker films due to the greater fringing field from the tip hemisphere compared to thinner films. At larger distances, however, the electric field in thicker films approaches the electric field of thinner film, becoming thickness independent. As discussed in the paper, the COMSOL simulation was used to further support using Eq. 5, the electric field due to the cone, for the electric field at lateral distances larger than about tip diameter.